In this paper a full description of order isomorphisms between effect algebras of atomic JBW-algebras is given. We will derive a closed formula for the order isomorphisms on the effect algebra of type I factors by proving that the invertible part of the effect algebra of a type I factor is left invariant. This yields an order isomorphism on the whole cone, for which a characterisation exists. Furthermore, we will show that the obtained formula for the order isomorphism on the invertible part can be extended to the whole effect algebra again. As atomic JBW-algebras are direct sums of type I factors and order isomorphisms factor through the direct sum decomposition, this yields the desired description.

中文翻译：

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原子JBW-代数阶区间的阶同构

在本文中，给出了原子JBW-代数的效应代数之间的阶同构的完整描述。我们将通过证明 I 型因子的效应代数的可逆部分保持不变来推导出 I 型因子效应代数的阶同构的封闭公式。这在整个锥体上产生了一个阶同构，为此存在一个表征。此外，我们将证明所得到的可逆部分阶同构的公式可以再次推广到整个效应代数。由于原子 JBW 代数是通过直接和分解的 I 类因子和阶同构因子的直接和，这产生了所需的描述。